Exponents Exercises
1. Simplify by using the rules of exponents.
(a) \( \displaystyle \frac{36 a^4 b^5}{100 a^7 b^3} \)
Solution
\(
\begin{aligned}
\frac{36 a^4 b^5}{100 a^7 b^3} &= \frac{9 \cdot 4}{25 \cdot 4} \cdot a^{4-7} \cdot b^{5-3} \\
&= \frac{9}{25} a^{-3} b^2 \\
&= \frac{9b^2}{25a^3}
\end{aligned}
\)
(b) \( \displaystyle \frac{27 a^2 b^5}{\left(9 a^2 b\right)^2} \)
Solution
\(
\begin{aligned}
\frac{27 a^2 b^5}{\left(9 a^2 b\right)^2} &= \frac{27 a^2 b^5}{81 a^4 b^2} \\
&= \frac{27}{81} a^{2-4} b^{5-2} \\
&= \frac{1}{3} a^{-2} b^3 \\
&= \frac{b^3}{3a^2}
\end{aligned}
\)
(c) \( \displaystyle \left(\frac{-135 a^4 b^5 c^6}{315 a^6 b^7 c^8}\right)^2 \)
Solution
\(
\begin{aligned}
\left(\frac{-135 a^4 b^5 c^6}{315 a^6 b^7 c^8}\right)^2 &= \left( \frac{-3 \cdot 45}{7 \cdot 45} a^{4-6} b^{5-7} c^{6-8} \right)^2 \\
&= \left( -\frac{3}{7} a^{-2} b^{-2} c^{-2} \right)^2 \\
&= \frac{9}{49} a^{-4} b^{-4} c^{-4} \\
&= \frac{9}{49a^4b^4c^4}
\end{aligned}
\)
(d) \( \displaystyle \left(\frac{x^4}{y^5}\right)^3\left(\frac{y^3}{x^2}\right)^2 \)
Solution
\(
\begin{aligned}
\left(\frac{x^4}{y^5}\right)^3\left(\frac{y^3}{x^2}\right)^2 &= \frac{x^{12}}{y^{15}} \cdot \frac{y^6}{x^4} \\
&= x^{12-4} y^{6-15} \\
&= x^8 y^{-9} \\
&= \frac{x^8}{y^9}
\end{aligned}
\)
(e) \( \displaystyle \frac{2^{3^2}}{\left(2^2\right)^3} \)
Solution
\(
\begin{aligned}
\frac{2^{3^2}}{\left(2^2\right)^3} &= \frac{2^9}{2^6} \\
&= 2^{9-6} \\
&= 2^3 \\
&= 8
\end{aligned}
\)
2. Evaluate the followings.
(a) \( \displaystyle \frac{54^2 \times 12^3 \times 64^2\left(3^2 \times 4^3 \times 5^2\right)^3}{\left(3^2 \times 15 \times 20^3\right)^4} \)
Solution
\(
\begin{aligned}
\text{Num} &= (2 \cdot 3^3)^2 \cdot (2^2 \cdot 3)^3 \cdot (2^6)^2 \cdot (3^2 \cdot 2^6 \cdot 5^2)^3 \\
&= (2^2 \cdot 3^6) \cdot (2^6 \cdot 3^3) \cdot 2^{12} \cdot (3^6 \cdot 2^{18} \cdot 5^6) \\
&= 2^{2+6+12+18} \cdot 3^{6+3+6} \cdot 5^6 \\
&= 2^{38} \cdot 3^{15} \cdot 5^6 \\
\text{Den} &= (3^2 \cdot (3 \cdot 5) \cdot (2^2 \cdot 5)^3)^4 \\
&= (3^2 \cdot 3 \cdot 5 \cdot 2^6 \cdot 5^3)^4 \\
&= (2^6 \cdot 3^3 \cdot 5^4)^4 \\
&= 2^{24} \cdot 3^{12} \cdot 5^{16} \\
\text{Exp} &= \frac{2^{38} \cdot 3^{15} \cdot 5^6}{2^{24} \cdot 3^{12} \cdot 5^{16}} = 2^{14} \cdot 3^3 \cdot 5^{-10} = \frac{27 \cdot 2^{14}}{5^{10}}
\end{aligned}
\)
(b) \( \displaystyle \left(\frac{343}{36}\right)^3\left(\frac{540}{56}\right)^4 \)
Solution
\(
\begin{aligned}
\text{Exp} &= \left(\frac{7^3}{2^2 \cdot 3^2}\right)^3 \left(\frac{2^2 \cdot 3^3 \cdot 5}{2^3 \cdot 7}\right)^4 \\
&= \frac{7^9}{2^6 \cdot 3^6} \cdot \left(\frac{3^3 \cdot 5}{2 \cdot 7}\right)^4 \\
&= \frac{7^9}{2^6 \cdot 3^6} \cdot \frac{3^{12} \cdot 5^4}{2^4 \cdot 7^4} \\
&= \frac{7^{9-4} \cdot 3^{12-6} \cdot 5^4}{2^{6+4}} \\
&= \frac{7^5 \cdot 3^6 \cdot 5^4}{2^{10}}
\end{aligned}
\)
(c) \( \displaystyle \left(\frac{33}{1056}\right)^3\left(\frac{768}{270}\right)^4\left(\frac{450}{48}\right)^3 \)
Solution
\(
\begin{aligned}
\text{Exp} &= \left(\frac{33}{33 \cdot 32}\right)^3 \left(\frac{256 \cdot 3}{27 \cdot 10}\right)^4 \left(\frac{150 \cdot 3}{16 \cdot 3}\right)^3 \\
&= \left(\frac{1}{2^5}\right)^3 \left(\frac{2^8 \cdot 3}{3^3 \cdot 2 \cdot 5}\right)^4 \left(\frac{2 \cdot 3 \cdot 5^2}{2^4}\right)^3 \\
&= (2^{-15}) \cdot \left(\frac{2^7}{3^2 \cdot 5}\right)^4 \cdot \left(\frac{3 \cdot 5^2}{2^3}\right)^3 \\
&= 2^{-15} \cdot \frac{2^{28}}{3^8 \cdot 5^4} \cdot \frac{3^3 \cdot 5^6}{2^9} \\
&= 2^{-15+28-9} \cdot 3^{3-8} \cdot 5^{6-4} \\
&= 2^4 \cdot 3^{-5} \cdot 5^2 \\
&= \frac{16 \cdot 25}{243} = \frac{400}{243}
\end{aligned}
\)
3. Simplify.
(a) \( \displaystyle \left(\frac{3^m}{15^n}\right)^3\left(\frac{45^n}{255^m}\right)^2 \)
Solution
\(
\begin{aligned}
\text{Exp} &= \frac{3^{3m}}{(3 \cdot 5)^{3n}} \cdot \frac{(3^2 \cdot 5)^{2n}}{(3 \cdot 5 \cdot 17)^{2m}} \\
&= \frac{3^{3m}}{3^{3n} \cdot 5^{3n}} \cdot \frac{3^{4n} \cdot 5^{2n}}{3^{2m} \cdot 5^{2m} \cdot 17^{2m}} \\
&= 3^{3m-3n+4n-2m} \cdot 5^{2n-3n-2m} \cdot 17^{-2m} \\
&= 3^{m+n} \cdot 5^{-n-2m} \cdot 17^{-2m} \\
&= \frac{3^{m+n}}{5^{2m+n} \cdot 17^{2m}}
\end{aligned}
\)
(b) \( \displaystyle \left(\frac{20^x}{400^y}\right)^2\left(\frac{150^{y^2}}{180^x}\right)^3 \)
Solution
\(
\begin{aligned}
\text{Exp} &= \frac{(2^2 \cdot 5)^{2x}}{((2^4 \cdot 5^2)^y)^2} \cdot \frac{((2 \cdot 3 \cdot 5^2)^{y^2})^3}{((2^2 \cdot 3^2 \cdot 5)^x)^3} \\
&= \frac{2^{4x} \cdot 5^{2x}}{2^{8y} \cdot 5^{4y}} \cdot \frac{2^{3y^2} \cdot 3^{3y^2} \cdot 5^{6y^2}}{2^{6x} \cdot 3^{6x} \cdot 5^{3x}} \\
&= 2^{4x-8y+3y^2-6x} \cdot 3^{3y^2-6x} \cdot 5^{2x-4y+6y^2-3x} \\
&= 2^{3y^2-2x-8y} \cdot 3^{3y^2-6x} \cdot 5^{6y^2-x-4y}
\end{aligned}
\)
(c) \( \displaystyle \frac{\left(x^3-y^3\right)(x+y)}{\left(x^2-y^2\right)^3} \)
Solution
\(
\begin{aligned}
\frac{\left(x^3-y^3\right)(x+y)}{\left(x^2-y^2\right)^3} &= \frac{(x-y)(x^2+xy+y^2)(x+y)}{[(x-y)(x+y)]^3} \\
&= \frac{(x-y)(x+y)(x^2+xy+y^2)}{(x-y)^3(x+y)^3} \\
&= \frac{x^2+xy+y^2}{(x-y)^2(x+y)^2} \\
&= \frac{x^2+xy+y^2}{(x^2-y^2)^2}
\end{aligned}
\)
(d) \( \displaystyle \frac{\left(x^{a-b} x^{b-c}\right)^a\left(\frac{x^a}{x^c}\right)^c}{\left(x^a x^c\right)^a \div\left(x^{a+c}\right)^c} \)
Solution
\(
\begin{aligned}
\text{Exp} &= \frac{(x^{a-c})^a \cdot (x^{a-c})^c}{x^{a^2+ac} \div x^{ac+c^2}} \\
&= \frac{x^{a^2-ac} \cdot x^{ac-c^2}}{x^{a^2+ac-(ac+c^2)}} \\
&= \frac{x^{a^2-c^2}}{x^{a^2-c^2}} \\
&= 1
\end{aligned}
\)
4. Evaluate the followings.
(a) \( \displaystyle (-3)^{-2} \)
Solution
\(
\begin{aligned}
(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}
\end{aligned}
\)
(b) \( \displaystyle -3^{-3} \)
Solution
\(
\begin{aligned}
-3^{-3} = -\frac{1}{3^3} = -\frac{1}{27}
\end{aligned}
\)
(c) \( \displaystyle -2^0+5^{-1} \)
Solution
\(
\begin{aligned}
-2^0+5^{-1} = -1 + \frac{1}{5} = \frac{-5+1}{5} = -\frac{4}{5}
\end{aligned}
\)
(d) \( \displaystyle (-2)^{-3}+2^{-2}-2^{-4} \)
Solution
\(
\begin{aligned}
(-2)^{-3}+2^{-2}-2^{-4} &= \frac{1}{(-2)^3} + \frac{1}{2^2} - \frac{1}{2^4} \\
&= -\frac{1}{8} + \frac{1}{4} - \frac{1}{16} \\
&= \frac{-2 + 4 - 1}{16} \\
&= \frac{1}{16}
\end{aligned}
\)
(e) \( \displaystyle 5^0-(-3)^0 \)
Solution
\(
\begin{aligned}
5^0-(-3)^0 = 1 - 1 = 0
\end{aligned}
\)
(f) \( \displaystyle \frac{27^{-6}}{125^{-3}} \div \frac{9^{-2}}{25^{-4}} \)
Solution
\(
\begin{aligned}
\frac{(3^3)^{-6}}{(5^3)^{-3}} \div \frac{(3^2)^{-2}}{(5^2)^{-4}} &= \frac{3^{-18}}{5^{-9}} \cdot \frac{5^{-8}}{3^{-4}} \\
&= 3^{-18 - (-4)} \cdot 5^{-8 - (-9)} \\
&= 3^{-14} \cdot 5^1 \\
&= \frac{5}{3^{14}}
\end{aligned}
\)
(g) \( \displaystyle (-5)^0-(-5)^{-1}-(-5)^{-2}-(-5)^{-3} \)
Solution
\(
\begin{aligned}
&= 1 - \frac{1}{-5} - \frac{1}{(-5)^2} - \frac{1}{(-5)^3} \\
&= 1 + \frac{1}{5} - \frac{1}{25} + \frac{1}{125} \\
&= \frac{125 + 25 - 5 + 1}{125} \\
&= \frac{146}{125}
\end{aligned}
\)
(h) \( \displaystyle (-1)^{(-1)^{-1}} \)
Solution
\(
\begin{aligned}
(-1)^{-1} &= \frac{1}{-1} = -1 \\
(-1)^{-1} \implies (-1)^{-1} &= -1
\end{aligned}
\)
(i) \( \displaystyle \frac{\left(180^2\right)^{-3}\left(6 \cdot 90^{-2}\right)^3}{\left(40^{-3}\right)^2 \cdot 25^{-2}} \)
Solution
\(
\begin{aligned}
\text{Exp} &= \frac{(2^2 \cdot 3^2 \cdot 5)^{-6} \cdot (2 \cdot 3)^3 \cdot (2 \cdot 3^2 \cdot 5)^{-6}}{(2^3 \cdot 5)^{-6} \cdot (5^2)^{-2}} \\
&= \frac{2^{-12} \cdot 3^{-12} \cdot 5^{-6} \cdot 2^3 \cdot 3^3 \cdot 2^{-6} \cdot 3^{-12} \cdot 5^{-6}}{2^{-18} \cdot 5^{-6} \cdot 5^{-4}} \\
&= \frac{2^{-15} \cdot 3^{-21} \cdot 5^{-12}}{2^{-18} \cdot 5^{-10}} \\
&= 2^{-15+18} \cdot 3^{-21} \cdot 5^{-12+10} \\
&= 2^3 \cdot 3^{-21} \cdot 5^{-2} = \frac{8}{25 \cdot 3^{21}}
\end{aligned}
\)
(j) \( \displaystyle \frac{\left(2^{-3}-3^{-2}\right)^{-1}}{\left(2^{-3}+3^{-2}\right)^{-1}} \)
Solution
\(
\begin{aligned}
\frac{\left(\frac{1}{8} - \frac{1}{9}\right)^{-1}}{\left(\frac{1}{8} + \frac{1}{9}\right)^{-1}} &= \frac{\left(\frac{1}{72}\right)^{-1}}{\left(\frac{17}{72}\right)^{-1}} \\
&= \frac{72}{\frac{72}{17}} \\
&= 72 \cdot \frac{17}{72} = 17
\end{aligned}
\)
5. Simplify the followings.
(a) \( \displaystyle \left(-3 a^4\right)\left(4 a^{-7}\right) \)
Solution
\(
\begin{aligned}
\left(-3 a^4\right)\left(4 a^{-7}\right) &= -12 a^{4-7} \\
&= -12a^{-3} \\
&= -\frac{12}{a^3}
\end{aligned}
\)
(b) \( \displaystyle \left(\frac{2 x^{-4}}{5 y^2 z^3}\right)^{-2} \)
Solution
\(
\begin{aligned}
\left(\frac{2 x^{-4}}{5 y^2 z^3}\right)^{-2} &= \left(\frac{5 y^2 z^3}{2 x^{-4}}\right)^2 \\
&= \frac{25 y^4 z^6}{4 x^{-8}} \\
&= \frac{25 x^8 y^4 z^6}{4}
\end{aligned}
\)
(c) \( \displaystyle \left(\frac{x^{2 m+n} x^{3(m-n)}}{x^{m-2 n} x^{2 m-n}}\right)^{-3} \)
Solution
\(
\begin{aligned}
\text{Exp} &= \left( \frac{x^{2m+n+3m-3n}}{x^{m-2n+2m-n}} \right)^{-3} \\
&= \left( \frac{x^{5m-2n}}{x^{3m-3n}} \right)^{-3} \\
&= (x^{5m-2n-(3m-3n)})^{-3} \\
&= (x^{2m+n})^{-3} \\
&= x^{-6m-3n} = \frac{1}{x^{6m+3n}}
\end{aligned}
\)
(d) \( \displaystyle \left(\frac{2 x^{-3} y^2}{3^{-1} y^3}\right)^2\left(\frac{4 x^{-2} y^3}{3 x^5}\right)^3 \div\left(\frac{81 x^{-2}}{y^{-3}}\right)^{-2} \)
Solution
\(
\begin{aligned}
\text{Exp} &= (6 x^{-3} y^{-1})^2 \cdot \left(\frac{4 y^3}{3 x^7}\right)^3 \div (81 x^{-2} y^3)^{-2} \\
&= (36 x^{-6} y^{-2}) \cdot \left(\frac{64 y^9}{27 x^{21}}\right) \cdot (81 x^{-2} y^3)^2 \\
&= \left(\frac{36 \cdot 64}{27} x^{-27} y^7\right) \cdot (6561 x^{-4} y^6) \\
&= \left(\frac{256}{3} x^{-27} y^7\right) \cdot (6561 x^{-4} y^6) \\
&= 559872 x^{-31} y^{13} = \frac{559872 y^{13}}{x^{31}}
\end{aligned}
\)
(e) \( \displaystyle \frac{2 x+y}{x^{-1}+2 y^{-1}} \)
Solution
\(
\begin{aligned}
\frac{2 x+y}{x^{-1}+2 y^{-1}} &= \frac{2x+y}{\frac{1}{x} + \frac{2}{y}} \\
&= \frac{2x+y}{\frac{y+2x}{xy}} \\
&= (2x+y) \cdot \frac{xy}{2x+y} = xy
\end{aligned}
\)
(f) \( \displaystyle \left(x^{-2}-y^{-1}\right)^{-3} \)
Solution
\(
\begin{aligned}
\left(x^{-2}-y^{-1}\right)^{-3} &= \left( \frac{1}{x^2} - \frac{1}{y} \right)^{-3} \\
&= \left( \frac{y-x^2}{x^2 y} \right)^{-3} \\
&= \left( \frac{x^2 y}{y-x^2} \right)^3 = \frac{x^6 y^3}{(y-x^2)^3}
\end{aligned}
\)
(g) \( \displaystyle \frac{x^{-2}-y^{-2}}{x^{-1}+y^{-1}} \)
Solution
\(
\begin{aligned}
\frac{x^{-2}-y^{-2}}{x^{-1}+y^{-1}} &= \frac{(x^{-1}-y^{-1})(x^{-1}+y^{-1})}{x^{-1}+y^{-1}} \\
&= x^{-1}-y^{-1} \\
&= \frac{1}{x} - \frac{1}{y} = \frac{y-x}{xy}
\end{aligned}
\)
(h) \( \displaystyle \frac{\left(x+y^{-1}\right)^2}{1+x^{-1} y^{-1}} \)
Solution
\(
\begin{aligned}
\frac{\left(x+y^{-1}\right)^2}{1+x^{-1} y^{-1}} &= \frac{\left(x+\frac{1}{y}\right)^2}{1+\frac{1}{xy}} \\
&= \frac{\left(\frac{xy+1}{y}\right)^2}{\frac{xy+1}{xy}} \\
&= \frac{(xy+1)^2}{y^2} \cdot \frac{xy}{xy+1} = \frac{x(xy+1)}{y}
\end{aligned}
\)
6. Simplify the following.
(a) \( \displaystyle \frac{y\left(2 x^4 y^2\right)^2}{2 x^4 y^0} \)
Solution
\(
\begin{aligned}
\frac{y\left(2 x^4 y^2\right)^2}{2 x^4 y^0} &= \frac{y(4 x^8 y^4)}{2 x^4 \cdot 1} \\
&= \frac{4 x^8 y^5}{2 x^4} \\
&= 2 x^4 y^5
\end{aligned}
\)
(b) \( \displaystyle \frac{b^{-1}}{\left(2 a^4 b^0\right)^0 \cdot 2 a^{-3} b^2} \)
Solution
\(
\begin{aligned}
\frac{b^{-1}}{\left(2 a^4 b^0\right)^0 \cdot 2 a^{-3} b^2} &= \frac{b^{-1}}{1 \cdot 2 a^{-3} b^2} \\
&= \frac{1}{2} a^3 b^{-1-2} \\
&= \frac{1}{2} a^3 b^{-3} = \frac{a^3}{2b^3}
\end{aligned}
\)
(c) \( \displaystyle \frac{2 y z x^2}{2 x^4 y^4 z^{-2} \cdot\left(z y^2\right)^4} \)
Solution
\(
\begin{aligned}
\frac{2 y z x^2}{2 x^4 y^4 z^{-2} \cdot\left(z y^2\right)^4} &= \frac{2 x^2 y z}{2 x^4 y^4 z^{-2} \cdot z^4 y^8} \\
&= \frac{x^2 y z}{x^4 y^{12} z^2} \\
&= x^{-2} y^{-11} z^{-1} = \frac{1}{x^2 y^{11} z}
\end{aligned}
\)
(d) \( \displaystyle \frac{2 b^4 c^{-2} \cdot\left(2 b^3 c^2\right)^{-4}}{a^{-2} b^4} \)
Solution
\(
\begin{aligned}
\frac{2 b^4 c^{-2} \cdot\left(2 b^3 c^2\right)^{-4}}{a^{-2} b^4} &= \frac{2 b^4 c^{-2} \cdot 2^{-4} b^{-12} c^{-8}}{a^{-2} b^4} \\
&= \frac{2^{-3} b^{-8} c^{-10}}{a^{-2} b^4} \\
&= 2^{-3} a^2 b^{-12} c^{-10} = \frac{a^2}{8 b^{12} c^{10}}
\end{aligned}
\)
(e) \( \displaystyle \frac{2 k h^0 \cdot 2 h^{-3} k^0}{\left(2 k j^3\right)^2} \)
Solution
\(
\begin{aligned}
\frac{2 k h^0 \cdot 2 h^{-3} k^0}{\left(2 k j^3\right)^2} &= \frac{(2k)(2h^{-3})}{4 k^2 j^6} \\
&= \frac{4 k h^{-3}}{4 k^2 j^6} \\
&= k^{-1} h^{-3} j^{-6} = \frac{1}{k h^3 j^6}
\end{aligned}
\)
(f) \( \displaystyle \left(\frac{\left(2 x^{-3} y^0 z^{-1}\right)^3 \cdot x^{-3} y^2}{2 x^3}\right)^{-2} \)
Solution
\(
\begin{aligned}
\text{Exp} &= \left( \frac{(8 x^{-9} z^{-3})(x^{-3} y^2)}{2 x^3} \right)^{-2} \\
&= \left( \frac{8 x^{-12} y^2 z^{-3}}{2 x^3} \right)^{-2} \\
&= (4 x^{-15} y^2 z^{-3})^{-2} \\
&= 4^{-2} x^{30} y^{-4} z^6 = \frac{x^{30} z^6}{16 y^4}
\end{aligned}
\)
(g) \( \displaystyle \frac{\left(c b^3\right)^2 \cdot 2 a^{-3} b^2}{\left(a^3 b^{-2} c^3\right)^3} \)
Solution
\(
\begin{aligned}
\frac{\left(c b^3\right)^2 \cdot 2 a^{-3} b^2}{\left(a^3 b^{-2} c^3\right)^3} &= \frac{c^2 b^6 \cdot 2 a^{-3} b^2}{a^9 b^{-6} c^9} \\
&= \frac{2 a^{-3} b^8 c^2}{a^9 b^{-6} c^9} \\
&= 2 a^{-12} b^{14} c^{-7} = \frac{2 b^{14}}{a^{12} c^7}
\end{aligned}
\)
(h) \( \displaystyle \frac{2 q^4 \cdot m^2 p^2 q^4}{\left(2 m^{-4} p^2\right)^3} \)
Solution
\(
\begin{aligned}
\frac{2 q^4 \cdot m^2 p^2 q^4}{\left(2 m^{-4} p^2\right)^3} &= \frac{2 m^2 p^2 q^8}{8 m^{-12} p^6} \\
&= \frac{1}{4} m^{14} p^{-4} q^8 \\
&= \frac{m^{14} q^8}{4 p^4}
\end{aligned}
\)
(i) \( \displaystyle \frac{\left(y x^{-4} z^2\right)^{-1}}{z^3 \cdot x^2 y^3 z^{-1}} \)
Solution
\(
\begin{aligned}
\frac{\left(y x^{-4} z^2\right)^{-1}}{z^3 \cdot x^2 y^3 z^{-1}} &= \frac{y^{-1} x^4 z^{-2}}{x^2 y^3 z^2} \\
&= x^2 y^{-4} z^{-4} \\
&= \frac{x^2}{y^4 z^4}
\end{aligned}
\)
(j) \( \displaystyle \frac{2 m p n^{-3}}{\left(m^0 n^{-4} p^2\right)^3 \cdot 2 n^2 p^0} \)
Solution
\(
\begin{aligned}
\frac{2 m p n^{-3}}{\left(m^0 n^{-4} p^2\right)^3 \cdot 2 n^2 p^0} &= \frac{2 m p n^{-3}}{(n^{-12} p^6) \cdot (2 n^2)} \\
&= \frac{2 m p n^{-3}}{2 n^{-10} p^6} \\
&= m p^{-5} n^7 \\
&= \frac{m n^7}{p^5}
\end{aligned}
\)
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